Upper bounds for random multiplicative functions

In this question let $f$ denote a Rademacher or Steinhaus random multiplicative function. It was conjectured by A. J. Harper (https://arxiv.org/pdf/1703.06654) that $\sum_{n\le x}f(n)=O(x^{1/2}(\log\log x)^{1/4+\epsilon}).$ Harper proved the lower bound of this order of magnitude, and his conjecture is essentially that this bound is sharp.

A recent breakthrough of Durkan and Pearce-Crump (https://arxiv.org/pdf/2607.29429) shows that this conjecture is true (thus disproving a conjecture of Erdos).

I am new to this particular area but want to understand the proof and how the authors managed to get the sharp bound - as I understand it, this result demonstrates that $f$ has significant internal cancellation, in particular better than the bound one would obtain if all of the $f$ terms were independent. So my question is

How exactly is the sharp bound obtained here? What is the new method introduced here/can it be used to approach other problems in the area?

I also understand that one often uses such models to model arithmetic functions such as $\mu(n)$; does this bound tell us anything even heuristically about $\sum_{n\le x}\mu(n)$?